Begin2.DVI

I7-17. Vector equation of plane is(~r−~r 0 )·N~= 0where~r=xeˆ 1 +yˆe 2 +zeˆ 3 is variable point in plane,~r 0 is fixed point in ...

I7-26. 2 π I7-27. 19 24 I7-28. π I7-30. If~r=~r(u,v), thend~r=∂~r∂udu+∂~r∂vdvand ds^2 =d~r·d~r= ∂~r ∂u· ∂~r ∂u(du) (^2) + 2∂~r ∂ ...

I7-34. (a)x,y-plane, coordinate curves~r(u 0 ,v)vertical lines,~r(u,v 0 )horizontal lines (b) x,y-plane polar coordinates,~r(u 0 ...

I7-38. (continued) Method II Use spherical coordinatesx= 3 sinθcosφ, y= 3 sinθsinφ, z= 3 cosθ dS=3dθ3 sinθdφ, F~·eˆn= x^2 +y^2 3 ...

I7-41. F=ω+λ 1 g+λ 2 h=x^2 +y^2 +z^2 +λ 1 g+λ 2 h ∂F ∂λ 1 =g=x+y+z−6 = 0 ∂F ∂λ 2 =h= 3x+ 5y+ 7z−34 = 0 ∂F ∂x =2x+λ 1 + 3λ 2 = 0 ...

I7-45. ~rc= ∑n j=1 mj~rj ∑n j=1 mj Note that this is a weighted sum of the vectors~ri where the weight factors arem 1 ,m 2 ,..., ...

Chapter 8 I8-1. I8-2. (i) gradφ= 4ˆe 1 − 3 ˆe 2 (iii)gradφ= 2xˆe 1 + 2yˆe 2 (ii) I8-3. (iii)z=x^2 +y^2 paraboloid N~ =− 2 xˆe 1 ...

I8-6. (i)r= √ x^2 +y^2 +z^2 and ∇rn= ∂rn ∂x eˆ 1 +∂r n ∂y ˆe 2 +∂r n ∂z ˆe 3 =nrn−^1 [ ∂r ∂x ˆe 1 +∂r ∂y ˆe 2 +∂r ∂z eˆ 3 ] =nrn ...

I8-13. V =^1 2 kx^2 withF~ =mddt^2 ~r 2. UseF~ = gradV =−kxˆe 1. That is, if the spring is stretched in the positive direction a ...

I8-19. IfV~=∇φ, thendiv~V =∇^2 φ= 0andcurlV~= curl (∇φ) =~ 0 I8-21. Only flux is from top surface and ∫∫ S F~·dS~ = 16π and from ...

I8-32. If origin outside ofS, then use the divergence theorem and show ∫∫ S ˆen·~r r^3 dS= ∫∫∫ V ∇· ( ~r r^3 ) dV and then show∇ ...

I8-52. IfA~=αE~ 1 +βE~ 2 +γE~ 3 andE~ 1 ·E~ 2 = 0,E~ 1 ·E~ 3 = 0,E~ 2 ·E~ 3 = 0, then show A~·E~ 1 =αE~ 1 ·E~ 1 or α= A~·E~^1 E~ ...

Chapter 9 I9-1. U=U(x) =c 1 x+c 2 , U=U(r) =c 1 lnr+c 2 , U=U(ρ) =−c 1 /ρ+c 2 I9-2. Streamlinessinαx+ cosαy=cand velocity field ...

I9-9. Integrate the equations ∂φ∂x=x 2 −+kyy 2 and∂φ∂y=x 2 kx+y 2 to obtain φ=−ktan−^1 ( x y ) +C 1 and φ=ktan−^1 (y x ) +C 2 Sh ...

I9-13. (b)~V = [−(x−y)z−yz 0 +x 0 z 0 ]ˆe 2 I9-14. φ=−mgz, W= ∫h 2 h 1 F~·d~r= ∫h 2 h 1 −mgdz=−mg(h 2 −h 1 ) =φ(h 2 )−φ(h 1 ) I9 ...

I9-24. |~r|= (x^2 +y^2 +z^2 )^1 /^2 = (~r·~r)^1 /^2 so that|~r|ν= (x^2 +y^2 +z^2 )ν/^2 =φ ∂φ ∂x =ν 2 (x^2 +y^2 +z^2 )ν/^2 −^1 (2 ...

Chapter 10 I10-1. (c) AB= [ 36 59 11 18 ] (d) B−^1 A−^1 = [ −18 59 11 − 36 ] I10-2. AA−^1 =I (AA−^1 )T=IT=I (A−^1 )TAT=I multipl ...

I10-10. (a) IfAT=AandBT=B, then(AB)T=BTAT=BA=AB (b) If((AB)T= (AB), thenBTAT=ABwhich impliesBA=AB I10-12. IfAB=BA, then one must ...

I10-28. A−^1 =   1 0 0 −3 1 0 11 −5 1   B−^1 =   1 2 − 7 − 8 0 1 − 3 − 3 0 0 1 1 0 0 0 1   C−^1 =   8 −6 4 − 2 −6 1 ...

Chapter 11 I11-1. (a) 10/15 (b) 3/15 (c) 2/15. I11-2. (a) (10/15)(3/14) (b) (10/15)(9/14) (c) (2/15)(3/14) I11-3. Method I Label ...